The midfielder has yet to agree terms with the Midlands club but is due to hold further talks with Dan Ashworth, West Brom's sporting director, after an encouraging initial meeting.
"Dan has had further talks with him and the talks have been relatively positive," said Hodgson.
"We believe he's capable of playing top-flight football again and we're very interested in him."
The 30-year-old passed a minor medical at the Hawthorns but will have to undergo a more rigorous test after being plagued by knee injuries in recent years.
"If the day ever came when he was even close to signing a contract, he'd have to go through a more rigorous medical, but I quite firmly believe that he would just as easily pass that," Hodgson added.
Hargreaves was released by Manchester United at the end of the 2010/11 season after making just 39 appearances in his four years at Old Trafford.